Escape rates and physically relevant measures for billiards with small holes

Transcription

1 Fairfield University Mathematics Faculty Publications Mathematics Department Escape rates and physically relevant measures for billiards with small holes Mark Demers Fairfield University, Paul Wright Lai-Sang Young Copyright 2010 Springer-Verlag Peer Reviewed Repository Citation Demers, Mark; Wright, Paul; and Young, Lai-Sang, "Escape rates and physically relevant measures for billiards with small holes" (2010). Mathematics Faculty Publications Published Citation Mark Demers, Paul Wright and Lai-Sang Young, "Escape rates and physically relevant measures for billiards with small holes," Communications in Mathematical Physics 294: 2 (2010), This Article is brought to you for free and open access by the Mathematics Department at DigitalCommons@Fairfield. It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of DigitalCommons@Fairfield. For more information, please contact digitalcommons@fairfield.edu.

2 Escape Rates and Physically Relevant Measures for Billiards with Small Holes Mark Demers Paul Wright Lai-Sang Young January 21, 2010 Abstract We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map. This paper is about leaky dynamical systems, or dynamical systems with holes. Consider a dynamical system defined by a map or a flow on a phase space M, and let H M be a hole through which orbits escape, that is to say, once an orbit enters H, we stop considering it from that point on. Starting from an initial probability distribution µ 0 on M, mass will leak out of the system as it evolves. Let µ n denote the distribution remaining at time n. The most basic question one can ask about a leaky system is its rate of escape, i.e. whether µ n (M) ϑ n for some ϑ. Another important question concerns the nature of the remaining distribution. One way to formulate that is to normalize µ n, and to inquire about properties of µ n /µ n (M) as n tends to infinity. Such limiting distributions, when they exist, are not invariant; they are conditionally invariant, meaning they are invariant up to a normalization. Comparisons of systems with small holes with the corresponding closed systems, i.e. systems for which the holes have been plugged, are also natural. These are some of the questions we will address in this paper. Department of Mathematics and Computer Science, Fairfield University. mdemers@fairfield.edu. This research is partially supported by NSF grant DMS Department of Mathematics, University of Maryland. paulrite@math.umd.edu. This research is partially supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. This author would also like to thank The Courant Institute of Mathematical Sciences, New York University, where he was affiliated when this project began. Courant Institute of Mathematical Sciences, New York University. lsy@cims.nyu.edu. This research is partially supported by a grant from the NSF. The authors would like to thank MSRI, Berkeley, and ESI, Vienna, where part of this work was carried out. 1

3 We do not consider these questions in the abstract, however; for a review paper in this direction, see [DY]. Our context here is that of billiard systems with small holes. Specifically, we carry out our analysis for the collision map of a 2-dimensional periodic Lorentz gas, and expect our results to be extendable to other dispersing billiards. Our holes are physical holes, in the sense that they are derived from holes in the physical domain of the system, i.e., the billiard table: we consider both convex holes away from the scatterers and holes that live on the boundaries of the scatterers. The holes considered in this paper are very small, but their placements are immaterial. For these leaky systems, we prove that there is a common rate of escape and a common limiting distribution for a large class of natural initial distributions including those with densities with respect to Liouville measure. These conditionally invariant measures, therefore, can be viewed as characteristic of the leaky systems in question, in a way that is analogous to physical measures or SRB measures for closed systems. We show, in fact, that as hole size tends to zero, these measures tend to the natural invariant measure of the corresponding closed billiard system. Our proof involves constructing a Markov tower extension with a special property over the billiard map, the new requirement being that it respects the hole. Let us backtrack a little for readers not already familiar with these ideas: In much the same way that Markov partitions have proved to be very useful in the study of Anosov and Axiom A diffeomorphisms, it was shown, beginning with [Y] and continued in a number of other papers, that many systems with sufficiently strong hyperbolic properties (but which are not necessarily uniformly hyperbolic) admit countable Markov extensions. Roughly speaking, these extensions behave like countable state Markov chains with nonlinearity ; they have considerably simpler structures than the original dynamical system. The idea behind this work is that escape dynamics are much simpler in a Markov setting when the hole corresponds to a collection of states ; this is what we mean by the Markov extension respecting the hole. All this is not for free, however. We pay a price with a somewhat elaborate construction of the tower, and again when we pass the information back to the billiard system, in exchange for having a Markov structure to work with in the treatment of the hole. There are advantages to this route of proof: First, once a Markov extension is constructed for a system, it can be used many times over for entirely different purposes. For the billiard maps studied here, these extensions were constructed in [Y]; our main task is to adapt them to holes. Second, once results on escape dynamics are established on towers, they apply to all Markov extensions. Here, the desired results are already known in a special case, namely expanding towers [BDM]; we need to extend them to the general, hyperbolic setting. What we propose here is a unified, generic approach for dealing with holes in dynamical systems, one that can, in principle, be carried out for all systems that admit Markov towers. Such systems include logistic maps, rank one attractors including the Hénon family, piecewise hyperbolic maps and other dispersing billiards in 2 or more dimensions. Conditionally invariant measures were first introduced in probabilistic settings, namely countable state Markov chains and topological Markov chains, beginning with [V] and more recently in [FKMP] and [CMS3]. In this setting, such measures are called quasi-stationary distributions and the existence of a Yaglom limit corresponds to the limit µ n /µ n (M), which we use here to identify a physical conditionally invariant measure for the leaky system. The first works to study deterministic systems with holes took advantage of finite Markov partitions. These include: Expanding maps on R n with holes which are elements of a finite 2

4 Markov partition [PY, CMS1, CMS2]; Smale horseshoes [C1, C2]; Anosov diffeomorphisms [CM1, CM2, CMT1, CMT2]; billiards with convex scatterers satisfying a non-eclipsing condition [LM, R] and large parameter logistic maps whose critical point maps out of the interval [HY]. In the latter two, the holes are chosen in such a way that the surviving dynamics are uniformly expanding or hyperbolic with Markov partitions. First results which drop Markov requirements on the map include piecewise expanding maps of the interval [BaK, CV, LiM, D1, BDM]; Misiurewicz [D2] and Collet-Eckmann [BDM] maps with generic holes; and piecewise uniformly hyperbolic maps [DL]. The tower construction is used in the one-dimensional studies [D1, D2, BDM]. Typically a restriction on the size of the hole is introduced in order to control the dynamics when a finite Markov partition is absent. General conditions ensuring the existence of conditionally invariant measures are first given in [CMM]. The physical relevance of such measures, however, is unclear without further qualifications. As noted in [DY], under very weak assumptions on the dynamical system, many such measures exist: for any prescribed rate of escape, one can construct infinitely many conditionally invariant densities. This is the reason for the emphasis placed in this paper on the limit µ n /µ n (M), which identifies a unique, physically relevant conditionally invariant measure. This paper is organized as follows: Our results are formulated in Sect. 1. In Sects. 2 and 3, the geometry of billiard maps and holes are looked at carefully as we modify previous constructions to give a generalized horseshoe that respects the hole. Out of this horseshoe, a Markov tower extension is constructed and results on escape dynamics on it proved; this is carried out in Sects. 4 and 5. These results are passed back to the billiard system in Sect. 6, where the remaining theorems are also proved. 1 Formulation of Results 1.1 Basic definitions We consider a closed dynamical system defined by a self-map f of a manifold M, and let H M be a hole through which orbits escape, i.e., we stop considering an orbit once it enters H. In this paper we are primarily concerned with holes that are open subsets of the phase space; they are not too large and generally not f-invariant. We will refer to the triplet (f, M, H) as a leaky system. First we introduce some notation. Let M = M\H. At least to begin with, let us make a formal distinction between f and f = f ( M f 1 M) : M f 1 M M, and write f n = f n ( n i=0 f i M). Let η be a probability measure on M. We define f η to be the measure on M defined by ( f η)(a) = η( f 1 A) for each Borel set A M. If η is an initial distribution on M, then η (n) := f n η/ f n η is the normalized distribution of points remaining in M after n units of time. Given an initial distribution η, the most basic question is the rate at which mass is leaked out of the system. We define the escape rate starting from η to be log ϑ(η) where 1 log ϑ(η) = lim n n log η ( n ) f i M i=0 3 assuming such a limit exists.

5 Another basic object is the limiting distribution η ( ) defined to be η ( ) = lim n η (n) if this weak limit exists. Of particular interest is when there is a number ϑ and a probability measure µ with the property that for all η in a large class of natural initial distributions (such as those having densities with respect to Lebesgue measure), we have ϑ(η) = ϑ and η ( ) = µ. In such a situation, µ can be thought of as a physical measure for the leaky system (f, M, H), in analogy with the idea of physical measures for closed systems. A Borel probability measure η on M is said to be conditionally invariant if it satisfies f η = ϑη for some ϑ (0, 1]. Clearly, the escape rate of a conditionally invariant measure η is well defined and is equal to log ϑ. Most leaky dynamical systems admit many conditionally invariant measures; see [DY]. In particular, limiting distributions, when they exist, are often conditionally invariant; they are among the more important conditionally invariant measures from an observational point of view. Finally, when a physical measure η for a leaky system (f, M, H) has absolutely continuous conditional measures on the unstable manifolds of the underlying closed system (f, M), we will call it an SRB measure for the leaky system, in analogy with the idea of SRB measures for closed systems. 1.2 Setting of present work The underlying closed dynamical system here is the billiard map associated with a 2- dimensional periodic Lorentz gas. Let {Γ i : i = 1,, d} be pairwise disjoint C 3 simplyconnected curves on T 2 with strictly positive curvature, and consider the billiard flow on the table X = T 2 \ i {interiorγ i}. We assume the finite horizon condition, which imposes an upper bound on the number of consecutive tangential collisions with Γ i. The phase space of the unit-speed billiard flow is M = (X S 1 )/ with suitable identifications at the boundary. Let M = i Γ i [ π, π ] M be the cross-section to the billiard flow corresponding to collision with the scatterers, and let f : M M be the Poincaré map. The 2 2 coordinates on M are denoted by (r,ϕ) where r Γ i is parametrized by arc length and ϕ is the angle a unit tangent vector at r makes with the normal pointing into the domain X. We denote by ν the invariant probability measure induced on M by Liouville measure on M, i.e., dν = c cos ϕdrdϕ where c is the normalizing constant. We consider the following two types of holes: Holes of Type I. In the table X, a hole σ of this type is an open interval in the boundary of a scatterer. When q 0 Γ i, we refer to {q 0 } as an infinitesimal hole, and let Σ h (q 0 ) denote the collection of all open intervals σ Γ i in the h-neighborhood of q 0. A hole σ in X of this type corresponds to a set H σ M of the form (a, b) [ π 2, π 2 ]. Holes of Type II. A hole σ of this type is an open convex subset of X away from i Γ i and bounded by a C 3 simple closed curve with strictly positive curvature. As above, we regard {q 0 } X \ Γ i as an infinitesimal hole, and use Σ h (q 0 ) to denote the set of all σ in the h-neighborhood of q 0. In this case, σ X does not correspond directly to a set in M. Rather, σ corresponds directly to a set in M, the phase space for the billiard flow, and we must make a choice as to which set in the cross section M will represent the hole for the billiard map. There is a well defined set B σ M consisting of all (r,ϕ) whose trajectories under the billiard flow on M will enter σ S 1 before reaching M again. Thus H σ = f(b σ ) 4

6 is a natural candidate for the hole in M representing σ, and will be taken as such in this work. However, it would also have been possible to take B σ as the representative set. The geometry of B σ and H σ in phase space will be discussed in detail in Sect Also, we note that the requirement that σ be a C 3 simple closed curve with strictly positive curvature can be considerably relaxed. It is even possible to allow some holes σ that are not convex. See the remark at the end of Sect Statement of results Let G = G(H σ ) denote the set of finite Borel measures η on M that are absolutely continuous with respect to ν with dη/dν being (i) Lipschitz on each connected component of M and (ii) strictly positive on i=0f i M. Notice that measures on M with Lipschitz dη/dν correspond to measures on M having a Lipschitz density with respect to Liouville measure. Standing hypotheses for Theorems 1 3: We assume (1) f : M M is the billiard map defined in Sect. 1.2, (2) {q 0 } is an infinitesimal hole of either Type I or Type II, and (3) σ Σ h (q 0 ) where h > 0 is assumed to be sufficiently small. Theorem 1. (Common escape rate). All initial distributions η G have a common escape rate log ϑ for some ϑ < 1; more precisely, for all η G, ϑ(η) is well defined and is equal to ϑ. Theorem 2. (Common limiting distribution). (a) For all η G, the normalized surviving distributions f n η/ f n η converge weakly to a common conditionally invariant distribution µ with ϑ(µ ) = ϑ. (b) In fact, for all η G, there is a constant c(η) > 0 s.t. ϑ n f n η converges weakly to c(η)µ. Thus from an observational point of view, log ϑ is the escape rate and µ the physical measure for the leaky system (f, M, H σ ). Theorem 3. (Geometry of limiting distribution). (a) µ is singular with respect to ν; (b) µ has strictly positive conditional densities on local unstable manifolds. The precise meaning of the statement in part (b) of Theorem 3 is that there are countably many patches (V i, µ i ), i = 1, 2,..., where for each i, (i) V i M is the union of a continuous family of unstable curves {γ u }; (ii) µ i is a measure on V i whose conditional measures on {γ u } have strictly positive densities with respect to the Riemannian measures on γ u ; (iii) µ i µ for each i, and i µ i µ. This justifies viewing µ as the SRB measure for the leaky system (f, M, H σ ). Our final result can be interpreted as a kind of stability for the natural invariant measure ν of the billiard map without holes. 5

7 Theorem 4. (Small-hole limit). We assume (1) and (2) in the Standing Hypotheses above. Let σ h Σ h (q 0 ), h > 0, be an arbitrary family of holes, and let log ϑ (σ h ) and µ (σ h ) be the escape rate and physical measure for the leaky system (f, M, H σh ). Then ϑ (σ h ) 1 and µ (σ h ) ν as h 0. Some straightforward generalizations: Our proofs continue to hold under the more general conditions below, but we have elected not to discuss them (or to include them formally in the statement of our theorems) because keeping track of an increased number of objects will necessitate more cumbersome notation. 1. Holes. Our results apply to more general classes of holes than those described above. For example, we could fix a finite number of infinitesimal holes {q 0 },..., {q k } and consider σ = i σ i with σ i Σ h (q i ). In fact, we may take more than one σ i in each Σ h (q i ) for as long as the total number of holes is uniformly bounded. See Sect. 3.4 for further generalizations on the types of holes allowed. 2. Initial distributions. Theorems 1 and 2 (and consequently Theorems 3 and 4) remain true with G replaced by a broader class of measures. For example, we use only the Lipschitz property of dη/dν along unstable leaves, and it is sufficient for dη/dν to be strictly positive on large enough open sets (see Remark 6.3). Moreover, dη/dν need not be bounded provided it blows up sufficiently slowly near the singularity set for f. Finally, we remark that Theorem 2(b) continues to hold without requiring that dη/dν be strictly positive anywhere, except that now c(η) might be 0. 2 Relevant Dynamical Structures Our plan is to show that the billiard maps described in Sect. 1.2 admit certain structures called generalized horseshoes which can be arranged to respect the holes. The main results are summarized in Proposition 2.2 in Sect. 2.2 and proved in Sect Generalized horseshoes We begin by recalling the idea of a horseshoe with infinitely many branches and variable return times introduced in [Y] for general dynamical systems without holes. These objects will be referred to in this paper as generalized horseshoes. Following the notation in Sect. 1.1 of [Y], we consider a smooth or piecewise smooth invertible map f : M M, and let µ and µ γ denote respectively the Riemannian measure on M and on γ where γ M is a submanifold. We say the pair (Λ, R) defines a generalized horseshoe if (P1) (P5) below hold (see [Y] for precise formulation): (P1) Λ is a compact subset of M with a hyperbolic product structure, i.e., Λ = ( Γ u ) ( Γ s ) where Γ s and Γ u are continuous families of local stable and unstable manifolds, and µ γ {γ Λ} > 0 for every γ Γ u. (P2) R : Λ Z + is a return time function to Λ. Modulo a set of µ-measure zero, Λ is the disjoint union of s-subsets Λ j, j = 1, 2,, with the property that for each j, R Λj = R j Z + and f R j (Λ j ) is a u-subset of Λ. 6

8 There is a notion of separation time s 0 (, ), depending only on the unstable coordinate, defined for pairs of points in Λ, and there are numbers C > 0 and α < 1 such that the following hold for all x, y Λ: (P3) For y γ s (x), d(f n x, f n y) Cα n for all n 0. (P4) For y γ u (x) and 0 k n < s 0 (x, y), (a) d(f n x, f n y) Cα s 0(x,y) n ; (b) log Π n det Df u (f i x) i=k Cα s0(x,y) n. det Df u (f i y) (P5) (a) For y γ s (x), log Π det Df u (f i x) i=n Cα n for all n 0. det Df u (f i y) (b) For γ, γ Γ u, if Θ : γ Λ γ Λ is defined by Θ(x) = γ s (x) γ, then Θ is absolutely continuous and d(θ 1 µ γ ) dµ γ (x) = Π det Df u (f i x) i=0. det Df u (f i Θx) The meanings of the last three conditions are as follows: Orbits that have not separated are related by local hyperbolic estimates; they also have comparable derivatives. Specifically, (P3) and (P4)(a) are (nonuniform) hyperbolic conditions on orbits starting from Λ. (P4)(b) and (P5) treat more refined properties such as distortion and absolute continuity of Γ s, conditions that are known to hold for C 1+ε hyperbolic systems. We say the generalized horseshoe (Λ, R) has exponential return times if there exist C 0 > 0 and θ 0 > 0 such that for all γ Γ u, µ γ {R > n} C 0 θ n 0 for all n 0. The setting described above is that of [Y]; it does not involve holes. In this setting, we now identify a set H M (to be regarded later as the hole) and introduce a few relevant terminologies. Let (Λ, R) be a generalized horseshoe for f with Λ (M \ H). We say (Λ, R) respects H if for every i and every l with 0 l R i, f l (Λ i ) either does not intersect H or is completely contained in H. The following definitions of mixing are motivated by Markov-chain considerations: Let Λ s Λ be an s-subset. We say Λ s makes a full return to Λ at time n if there are numbers i 0, i 1,, i k with n = R i0 + + R ik such that Λ s Λ i0, f R i 0 + +R ij(λ s ) Λ ij+1 for j < k, and f n (Λ s ) is a u-subset of Λ. (i) We say the horseshoe (Λ, R) is mixing if there exists N such that for every n N, some s-subset Λ s (n) makes a full return at time n. (ii) If (Λ, R) respects H, then when we treat H as a hole, we say the surviving dynamics are mixing if in addition to the condition in (i), we require that f l Λ s (n) H = for all l with 0 l n. This is equivalent to requiring that Λ s (n) makes a full return to Λ at time n under the dynamics of f, where f is the map defined in Sect We note that the mixing of f in the usual sense of ergodic theory does not imply that any generalized horseshoe constructed is necessarily mixing in the sense of the last paragraph, nor does mixing of the horseshoe imply that of its surviving dynamics. 2.2 Main Proposition for billiards with holes With these general ideas out of the way, we now return to the setting of the present paper. From here on, f : M M is the billiard map of the 2-D Lorentz gas as in Sect The following result lies at the heart of the approach taken in this paper: 7

9 Proposition 2.1. (Theorem 6(a) of [Y]) The map f admits a generalized horseshoe with exponential return times. A few more definitions are needed before we are equipped to state our main proposition: We call Q M a rectangular region if Q = u Q s Q where u Q consists of two unstable curves and s Q two stable curves. We let Q(Λ) denote the smallest rectangular region containing Λ, and define µ u (Λ) := inf γ Γ u µ γ (Λ γ). Finally, for a generalized horseshoe (Λ, R) respecting a hole H, we define n(λ, R; H) = sup{n Z + : no point in Λ falls into H in the first n iterates}. In the rest of this paper, C and α will be the constants in (P3) (P5) for the closed system f. All notation is as in Sect Proposition 2.2. Given an infinitesimal hole {q 0 } of Type I or II, there exist C 0, κ > 0, θ 0 (0, 1), and a rectangular region Q such that for all small enough h we have the following: (a) For each σ Σ h (q 0 ), (i) f admits a generalized horseshoe (Λ (σ), R (σ) ) respecting H σ ; (ii) both (Λ (σ), R (σ) ) and the corresponding surviving dynamics are mixing. (b) All σ Σ h (q 0 ) have the following uniform properties: (i) Q(Λ (σ) ) Q 1, and µ u (Λ (σ) ) κ; (ii) µ γ {R (σ) > n} < C 0 θ n 0 for all n 0; (iii) (P3) (P5) hold with the constants C and α. Moreover, if n(h) = inf σ Σh (q 0 ) n(λ, R; H σ ), then n(h) as h 0. Clarification: 1. Here and in Sect. 3, there is a set, namely H σ, that is identified to be the hole, and a horseshoe is constructed to respect it. Notice that the construction is continued after a set enters H σ. For reasons to become clear in Sect. 6, we cannot simply disregard those parts of the phase space that lie in the forward images of H σ. 2. Proposition 2.2 treats only small h, i.e. small holes. The smallness of the holes and the uniformness of the estimates in part (b) are needed for the spectral arguments in Sect. 4 to apply. Without any restriction on h, all the conclusions of Proposition 2.2 remain true except for the following: (a)(ii), where for large holes the surviving dynamics need not be mixing, (b)(i), and (b)(ii), where C 0 and θ 0 may be σ-dependent. The assertions for large h will be evident from our proofs; no separate arguments will be provided. A proof of Proposition 2.2 will require that we repeat the construction in the proof of Proposition 2.1 and along the way, to carry out a treatment of holes and related issues. We believe it is more illuminating conceptually (and more efficient in terms of journal pages) to 1 By Q(Λ (σ) ) Q, we only wish to convey that both rectangular regions are located in roughly the same region of the phase space, M, and not anything technical in the sense of convergence. 8

10 focus on what is new rather than to provide a proof written from scratch. We will, therefore, proceed as follows: The rest of this section contains a review of all the arguments used in the proof of Proposition 2.1, with technical estimates omitted and specific references given in their place. A proof of Proposition 2.2 is given in Sect. 3. There we go through the same arguments point by point, explain where modifications are needed and treat new issues that arise. For readers willing to skip more technical aspects of the analysis not related to holes, we expect that they will get a clear idea of the proof from this paper alone. For readers who wish to see all detail, we ask that they read this proof alongside the papers referenced. 2.3 Outline of construction in [Y] In this subsection, the setting and notation are both identical to that in Sect. 8 of [Y]. Referring the reader to [Y] for detail, we identify below 7 main ideas that form the crux of the proof of Proposition 2.1. We will point out the use of billiard properties and other geometric facts that may potentially be impacted by the presence of holes. Holes are not discussed explicitly, however, until Sect. 3. Notation and conventions: In [Y], S 0 and M were used interchangeably. Here we use exclusively M. Clearly, f 1 M is the discontinuity set of f. (i) u- and s-curves. Invariant cones C u and C s are fixed at each point, and curves all of whose tangent vectors are in C u (resp. C s ) are called u-curves (resp. s-curves). (ii) The p-metric. Euclidean distance on M is denoted by d(, ). Unless declared otherwise, distances and derivatives along u- and s-curves are measured with respect to a semi-metric called the p-metric defined by cos ϕdr. These two metrics are related by cp(x, y) d(x, y) p(x, y) 1 2. By Wδ u (x), we refer to the piece of local unstable curve of p-length 2δ centered at x. (P3) (P5) in Sect. 2.1 hold with respect to the p-metric. See Sect. 8.3 in [Y] for details. (iii) Derivative bounds. With respect to the p-metric, there is a number λ > 1 so that all vectors in C u are expanded by λ and all vectors in C s contracted by λ 1. Furthermore, derivatives at x along u-curves are d(x, M) 1. For purposes of distortion control, homogeneity strips of the form { I k = (r,ϕ) : π 2 1 k < ϕ < π } 2 2 1, k k (k + 1) 2 0, are used, with {I k } defined similarly in a neighborhood of ϕ = π. For convenience, we 2 will refer to M \ ( k k0 I k ) as one of the I k. Important Geometric Facts ( ): The following facts are used many times in the proof: (a) the discontinuity set f 1 M is the union of a finite number of compact piecewise smooth decreasing curves, each of which stretches from {ϕ = π/2} to {ϕ = π/2}; (b) u-curves are uniformly transversal (with angles bounded away from zero) to M and to f 1 M. 1. Local stable and unstable manifolds. Only homogeneous local stable and unstable curves are considered. Homogeneity for W u δ, for example, means that for all n 0, f n W u δ 9

11 lies in no more than 3 contiguous I k. Let δ 1 > 0 be a small number to be chosen. We let λ 1 = λ 1 4, δ = δ 4 1, and define B + λ 1,δ 1 = {x M : d(f n x, M f 1 ( M)) δ 1 λ n 1 for all n 0}, B λ 1,δ 1 = {x M : d(f n x, M f( M)) δ 1 λ n 1 for all n 0}. We require d(f n x, f 1 ( M)) δ 1 λ n 1 to ensure the existence of a local unstable curve through x, while the requirement on d(f n x, M) is to ensure its homogeneity. 2 Similar reasons apply to stable curves. Observe that (i) for all x B + λ 1,δ 1, W10δ s (x) is well defined and homogeneous (this is straightforward since δ << δ 1 and λ 1 is closer to 1 than λ); and (ii) as δ 1 0, ν(b + λ 1,δ 1 ) 1 (this follows from a standard Borel-Cantelli type argument). Analogous statements hold for B λ 1,δ Construction of the Cantor set Λ. The choice of Λ is, in fact, quite arbitrary. We pick a density point x 1 of B + λ 1,2δ 1 B λ 1,2δ 1 at least 2δ 1 away from f 1 ( M) M f( M), and let Ω = W u δ (x 1). 3 For each n, we define Ω n = {y Ω : d(f i y, f 1 ( M)) δ 1 λ i 1 for 0 i n}, and let Ω = n Ω n. Then Ω B + λ 1,δ 1, by the footnote in item 1 above and our choice of x 1 far from M. Let Γ s consist of all Wδ s(y), y Ω, and let Γ u be the set of all homogeneous Wloc u curves that meet every γs Γ s and which extend by a distance > δ on both sides of the curves in Γ s. The set Λ, which is defined to be ( Γ u ) ( Γ s ), clearly has a hyperbolic product structure. (P5)(b) is standard. This together with the choice of x 1 guarantees µ γ {γ Λ} > 0 for γ Γ u, completing the proof of (P1). A natural definition of separation time for x, y γ u is as follows: Let [x, y] be the subsegment of γ u connecting x and y. Then f n x and f n y are not yet separated, i.e. s 0 (x, y) n, if for all i n, f i [x, y] is connected and is contained in at most 3 contiguous I k. With this definition of s 0 (, ), (P3) (P5)(a) are checked using previously known billiard estimates. 3. The return map f R : Λ Λ. We point out that there is some flexibility in choosing the return map f R : Certain conditions have to be met when a return takes place, but when these conditions are met, we are not obligated to call it a return; in particular, R is not necessarily the first time an s-subrectangle of Q u-crosses Q where Q = Q(Λ). We first define f R on Ω. Let Ω n = Ω n \ {R n}. On Ω n is a partition P n whose elements are segments representing distinct trajectories. The rules are different before and after a certain time R 1, a lower bound for which is determined by λ 1, δ 1 and the derivative of f. 4 2 In fact, provided δ 1 is chosen sufficiently small, one can verify that d(f n x, f 1 ( M)) δ 1 λ n 1 implies that d(f n+1 x, M) δ 1 λ (n+1) 1 for all n 0. This fact, which was not used in [Y], will be used in item 2 below to simplify our presentation. 3 Later we will impose one further technical condition on the choice of x 1. See the very end of Sect In [Y], properties of R 1 are used in 4 places: (I)(i) in Sect. 3.2, Sublemma 3 in Sect. 7.3, the paragraph following (**) in Sect. 8.4, and a requirement in Sect. 8.3 that stable manifolds pushed forward more than R 1 times are sufficiently contracted. 10

12 (a) For n < R 1, Pn is constructed from the results of the previous step 5 as follows: Let ω P n 1, and let ω be a component of ω Ω n. Inserting cut-points only where necessary, we divide ω into subsegments ω i with the property that f n (ω i ) is homogeneous. These are the elements of P n. No point returns before time R 1. (b) For n R 1, we proceed as in (a) to obtain ω i. If f n (ω i ) u-crosses the middle of Q with 1.5δ sticking out on each side, then we declare that R = n on ω i f n Λ, and the elements of P n ωi Ω n are the connected components of ω i \ f n Λ. Otherwise put ω i P n as before. This defines R on a subset of Ω (which we do not know yet has full measure); the definition is extended to the associated s-subset of Λ by making R constant on Wloc s -curves. The s-subsets associated with ω i f n Λ in (b) above are the Λ j in (P2). It remains to check that f R (Λ j ) is in fact a u-subset of Λ. This is called the matching of Cantor sets in [Y] and is a consequence of the fact that Ω is dynamically defined and that R 1 is chosen sufficiently large. It remains to prove that p{r n} decays exponentially with n. Paragraphs 4, 5 and 6 contain the 3 main ingredients of the proof, with the final count given in Growth of u-curves to long segments. This is probably the single most important point, so we include a few more details. We first give the main idea before adapting it to the form it is used. Let ε 0 > 0 be a number the significance of which we will explain later. Here we think of a u-curve whose p-length exceeds ε 0 > 0 as long. Consider a u-curve ω. We introduce a stopping time T on ω as follows. For n = 1, 2,, we divide f n ω into homogeneous segments representing distinguishable trajectories. For x ω, let T(x) = inf{n > 0 : the segment of f n ω containing f n x has p length > ε 0 }. Lemma 2.3. There exist D 1 > 0 and θ 1 < 1 such that for any u-curve ω, p(ω \ {T n}) < D 1 θ n 1 for all n 1. This lemma relies on the following important geometric property of the class of billiards in question. This choice of ε 0 > 0 is closely connected to this property: (*) ([BSC1], Lemma 8.4) The number of curves in n i=1f i ( M) passing through or ending in any one point in M is K 0 n, where K 0 is a constant depending only on the table X. Let α 0 := 2 1 k=k 0 where {I k 2 k, k k 0 } are the homogeneity strips, and assume that λ 1 + α 0 < 1. Choose m large enough that θ 1 := (K 0 m + 1) m(λ α 0 ) < 1. We may then fix ε 0 < δ to be small enough that every Wloc u -curve of p-length ε 0 has the property that it intersects K 0 m smooth segments of m 1 f i ( M), so that the f m -image of such a Wloc u -curve has (K 0m + 1) connected components. 5 In [Y], it was sufficient to allow returns to Λ at times that were multiples of a large fixed integer m. Not only is this not necessary (see Paragraph 4), here it is essential that we avoid such periodic behavior to ensure mixing. Thus we take m = 1 when choosing return times in Paragraph 3. This is the only substantial departure we make from the construction in [Y]. 11

13 The proof of Lemma 2.3, which follows [BSC2], goes as follows: Consider a large n, which we may assume is a multiple of m. (Once Lemma 2.3 is proved for multiples of m, the estimate can be extended to intermediate values by enlarging the constant D 1.) We label distinguishable trajectories by their I k -itineraries. Notice that because f i ω is the union of a number of (disconnected) u-curves, it is possible for many distinguishable trajectories to have the same I k -itinerary. Specifically, by (*), each trajectory of length jm, j Z +, gives birth to at most (K 0 m + 1) trajectories of length (j + 1)m with the same I k -itinerary. To estimate p(ω \ {T n}), we assume the worst case scenario, in which the f n -images of subsegments of ω corresponding to all distinguishable trajectories have length ε 0. We then sum over all possible itineraries using bounds on Df along u-curves in I k. We now adapt Lemma 2.3 to the form in which it will be used. Let ω = f k ω for some ω P k in the construction in Paragraph 3. As we continue to evolve ω, f n ω is not just chopped up by the discontinuity set, bits of it that go near f 1 ( M) will be lost by intersecting with f k+n Ω k+n, and we need to estimate p(ω n \ {T n}) where ω n := ω f k (Ω k+n ) takes into consideration these intersections and T is redefined accordingly. A priori this may require a larger bound than that given in Lemma 2.3: it is conceivable that there are segments that will grow to length ε 0 without losing these bits but which do not now reach this reference length. We claim that all such segments have been counted, because (i) the deletion procedure does not create new connected components; it merely trims the ends of segments adjacent to cut-points; and (ii) the combinatorics in Lemma 2.1 count all possible itineraries (and not just those that lead to short segments). This yields the desired estimate on p(ω n \ {T n}), which is Sublemma 2 in Sect. 8.4 of [Y]. 5. Growth of gaps of Λ. Let ω be the subsegment of some γ u Γ u connecting the two s-boundaries of Q. We think of this as a return in the construction outlined in Paragraph 3, with the connected components ω of ω c = ω \ Λ being f k -images of elements of P k. We define a stopping time T on ω c by considering one ω at a time and defining on it the stopping time in Paragraph 4. Lemma 2.4. There exist D 2 > 0 and θ 2 < 1 independent of ω such that p(ω c n \ {T n}) < D 2 θ n 2 for all n 1. The idea of the proof is as follows. We may identify ω with Ω (see Paragraph 2), so that the collection of ω is precisely the collection of gaps in Λ. We say ω is of generation q if this is the first time a part of ω is removed in the construction of Ω. There are two separate estimates: (I) := p(ω ); (II) := p(ω n\{t n}). q>εn gen(ω )=q q εn gen(ω )=q (I) has exponentially small p-measure: this follows from a comparison of the growth rate of Df along u-curves versus the rate at which these curves get cut (see Paragraph 4). (II) is bounded above by Cp(ω ) p(f q 1 ω ) D 1θ n q 1 1. q εn gen(ω )=q 12

14 This is obtained by applying the modified version of Lemma 2.3 to f q 1 ω. A lower bound on p(f q 1 ω ) can be estimated as these curves have not been cut by f 1 ( M) (though they may have been shortened to maintain homogeneity), reducing the estimate to q gen(ω )=q p(ω ), which is p(ω). 6. Return of long segments. This concerns the evolution of unstable curves after they have grown long, where long has the same meaning as in Paragraph 4. The following geometric fact from [BSC2] is used: (**) Given ε 0 > 0, n 0 s.t. for every homogeneous W u loc -curve ω with p(ω) > ε 0 and every q n 0, f q ω contains a homogeneous segment which u-crosses the middle half of Q with > 2δ sticking out from each side. We choose ε 0 > 0 as explained in Paragraph 4 above, and apply (**) with q = n 0 to the segments that arise in Paragraphs 4 and 5 when the stopping time T is reached. For example, ω here may be equal to f n ω where ω is a subsegment of the ω in the last paragraph of Paragraph 4 with T ω = n. We claim that a fixed fraction of such a segment will make a return within n 0 iterates. To guarantee that, two other facts need to be established: (i) The small bits deleted by intersecting with f n+k Ω n+k before the return still leave a segment which u-crosses the middle half of Q with > 1.5δ sticking out from each side; this is easily checked. (ii) For q n 0, (f q ) is uniformly bounded on f q -images of homogeneous segments that u-cross Q. This is true because a segment contained in I k for too large a k cannot grow to length δ in n 0 iterates. 7. Tail estimate of return time. We now prove p{r n} C 0 θ n 0 for some θ 0 < 1. On Ω, introduce a sequence of stopping times T 1 < T 2 < as follows: A stopping time T of the type in Paragraph 4 or 5 is initiated on a segment as soon as T k is reached, and T k+1 is set equal to T k +T. In this process, we stop considering points that are lost to deletions or have returned to Λ. The desired bound follows immediately from the following two estimates: (i) There exists ε > 0, D 3 1, and θ 3 < 1 such that p(t [ε n] > n) < D 3 θ n 3. (ii) There exists ε 1 > 0 such that if T k ω = n, then p(ω {R > n + n 0 }) (1 ε 1 )p(ω) where n 0 is as in (**) in Paragraph 6. (ii) is explained in Paragraph 6. To prove (i), we let p = [ε n], decompose Ω into sets of the form A(k 1,, k p ) = {x Ω : T 1 (x),, T p (x) are defined with T i = k i }, apply Lemmas 2.1 and 2.2 to each set and recombine the results. The argument here is combinatorial, and does not use further geometric information about the system. 2.4 Sketch of proof of (**) following [BSC2] Property (**) is a weaker version of Theorem 3.13 in [BSC2]. We refer the reader to [BSC2] for detail, but include an outline of its proof because a modified version of the argument will be needed in the proof of Proposition 2.2. We omit the proof of the following elementary fact, which relies on the geometry of the discontinuity set including Property (*): 13

15 Sublemma A. Given any u-curve γ, through µ γ -a.e. x γ passes a homogeneous W s δ(x) (x) for some δ(x) > 0. The analogous statement holds for s-curves. Instead of considering every Wloc u -curve as required in (**), the problem is reduced to a finite number of mixing boxes U 1, U 2,...,U k with the following properties: (i) U j is a hyperbolic product set defined by (homogeneous) families Γ u (U j ) and Γ s (U j ); located in the middle third of U j is an s-subset Ũj with ν(ũj) > 0; (ii) Γ u (U j ) fills up nearly 100% of the measure of Q(U j ); and (iii) every W u loc -curve ω with p(ω) > ε 0 passes through the middle third of one of the Q(U j ) in the manner shown in Fig. 1 (left). That (i) and (ii) can be arranged follows from Sublemma A. That a finite number of U j suffices for (iii) follows from a compactness argument. Next we choose a suitable subset Ũ0 Λ to be used in the mixing. To do that, first pick a hyperbolic product set U 0 related to Q(Λ) as shown in Fig. 1 (right). We require that it meet Q(Λ) in a set of positive measure, that it sticks out of Q(Λ) in the u-direction by more than 2δ, and that the curves in Γ u (U 0 ) fill up nearly 100% of Q(U 0 ). Let l 0 > 0 be a small number, and let Ũ0 U 0 consist of those density points of U 0 Q(Λ) with the additional property that if a homogeneous stable curve γ s with p(γ s ) < l 0 meets such a point, then p(γ s U 0 )/p(γ s ) 1. For l 0 small enough, ν(ũ0) > 0 because the foliation into W u loc -curves is absolutely continuous. Q(Λ) ω Q(U j ) Q(U 0 ) Figure 1: Left: A mixing box U j. Right: The target box U 0. By the mixing property of (f, ν), there exists n 0 such that for all q n 0, ν(f q (Ũj) Ũ0) > 0 for every Ũj. We may assume also that n 0 is so large that for q n 0, if x Ũj is such that f q x Ũ0, then p(f q (γ s (x))) < l 0 where γ s (x) is the stable curve in Γ s (Ũj) passing through x. Let q n 0 and j be fixed, and let x Ũj be as above. From the high density of unstable curves in both U j and U 0, we are guaranteed that there are two elements γ u 1,γ u 2 Γ u (U j ) sandwiching the middle third of Q(U j ) such that for each i, a subsegment of γ u i containing γ s (x) γ u i is mapped under f q onto some ˆγ u i Γ u (U 0 ). Let Q = Q (q, j) be the u- subrectangle of Q(U 0 ) with u Q = ˆγ u 1 ˆγ u 2. Sublemma B. f q Q is continuous, equivalently, Q ( q 0f i ( M)) =. Sublemma B is an immediate consequence of the geometry of the discontinuity set: By the choice of x 1 in item 2 of Sect. 2.3, Q M =. Suppose Q ( q 1f i ( M)). Since q 1f i ( M) is the union of finitely many piecewise smooth (increasing) u-curves each connected component of which stretches from {ϕ = π/2} to {ϕ = π/2}, and these curves cannot touch u Q, a piecewise smooth segment from q 1f i ( M) that enters Q through one 14

16 component of s Q must exit through the other. In particular, it must cross f q γ s (x), which is a contradiction. To prove (**), let ω be a W u loc -curve with p(ω) > ε 0. We pick U j so that ω passes through the middle third of U j as in (iii) above. Sublemma B then guarantees that f q (ω f q Q ) connects the two components of s Q. This completes the proof of (**), except that we have not yet verified that f q (ω f q Q ) is homogeneous. To finish this last point, we modify the above argument as follows: First, we define a Wloc u curve γ to be strictly homogeneous if for all n 0, f n γ is contained inside one homogeneity strip I k (n). Strict homogeneity for Wloc s curves is defined analogously. The conclusions of Sublemma A remain valid if, in its statement, the word homogeneous is replaced by strictly homogeneous. Thus the mixing boxes U 1,...,U k can be chosen so that their defining families are comprised entirely of strictly homogeneous local manifolds. Furthermore, if x 1 is also chosen as a density point of points with sufficiently long strictly homogeneous unstable curves, Γ u (U 0 ) can be chosen to be comprised entirely of strictly homogeneous W u loc - curves. Having done this, an argument very similar to the proof of Sublemma B shows that f i Q ( k I k ) = for 0 i q, and this completes the proof of (**). 3 Horseshoes Respecting Holes for Billiard Maps 3.1 Geometry of holes in phase space We summarize here some relevant geometric properties and explain how we plan to incorporate holes into our horseshoe construction. Holes of Type I. Recall from Sect. 1.2 that for q 0 Γ i and σ Σ h (q 0 ), H σ M is a rectangle of the form (a, b) [ π 2, π 2 ]. We define H σ := {a, b} [ π 2, π 2 ], i.e. H σ is the boundary of H σ viewed as a subset of M. It will also be convenient to let H 0 M denote the vertical line {q 0 } [ π 2, π 2 ]. To construct a horseshoe respecting H σ, it is necessary to view two nearby points as having separated when they lie on opposite sides of H σ or on opposite sides of H σ in M \ H σ. Thus it is convenient to view f 1 ( H σ ) as part of the discontinuity set of f. For simplicity, consider first the case where q 0 does not lie on a line in the table X tangent to more than one scatterer. Then f 1 ( H σ ) is a finite union of pairs of roughly parallel, smooth s-curves. (Recall that s-curves are negatively sloped, with slopes uniformly bounded away from 0 and.) Each of the curves comprising f 1 ( H σ ) begins and ends in M f 1 ( M), that is to say, the geometric properties of f 1 ( H σ ) f 1 ( M) are similar to those of f 1 ( M). Likewise, f( H σ ) is a finite of union of pairs of (increasing) u-curves that begin and end in M f( M), and it will be convenient to regard that as part of the discontinuity set of f 1. Let N ε ( ) denote the ε-neighborhood of a set. We will need the following lemma. Lemma 3.1. For each ε > 0 there is an h > 0 such that for each σ Σ h, H σ N ε (H 0 ), fh σ N ε (fh 0 ), and f 1 H σ N ε (f 1 H 0 ). As f is discontinuous, Lemma 3.1 is not immediate. However, it can be easily verified, and we leave the proof to the reader. 15

17 Points q 0 that lie on lines in X with multiple tangencies to scatterers lead to slightly more complicated geometries, and special care is needed when defining what is meant by fh 0 and f 1 H 0. For example, consider the case where q 0 Γ 3 lies on a line that is tangent to Γ 1 and Γ 2, but which is not tangent to any other scatterer including Γ 3. Suppose further that r 1 Γ 1, r 2 Γ 2 are the points of tangency, that r 2 is closer to q 0 than r 1 is, that no other scatterer touches the line segment [q 0, r 1 ], and that Γ 1 and Γ 2 both lie on the same side of [q 0, r 1 ]; see Fig. 2 (left). Let σ be a small hole of Type I with q 0 σ. Then in Γ 2 [ π/2, π/2], f 1 ( H σ ) appears as described above. However, Γ 2 obstructs the view of σ from Γ 1, and so in Γ 1 [ π/2, π/2], f 1 (H σ ) is a small triangular region whose three sides are composed of a segment from Γ 1 {π/2}, a segment from f 1 (Γ 2 {π/2}), and a single segment from f 1 ( H σ ). See Fig. 2 (right). As a consequence, when we write f 1 H 0, we include in this set not just (r 2, π/2), but also f 1 (r 2, π/2) = (r 1, π/2). This is necessary in order for Lemma 3.1 to continue to hold. Aside from such minor modifications, the case of multiple tangencies is no different than when they are not present, and we leave further details to the reader. f 1 (H σ ) Γ 3 + π 2 q 0 σ ϕ f 1 ( H σ ) f ( 1 Γ 2 { }) π 2 Γ 2 r 2 Γ 1 r 1 π 2 r 1 r Γ 1 Figure 2: An infinitesmal hole aligned with multiple tangencies. Left: q 0 lies on a line segment in the billiard table X that is tangent to two scatterers. Right: Induced singularity curves in the subset Γ 1 [ π/2, π/2] of the phase space M. Holes of Type II. For simplicity, consider first the case where q 0 does not lie on a line in the table X tangent to more than one scatterer. Recall from Sect. 1.2 that the hole H σ here is taken to be f(b σ ) where B σ consists of points in M which enter σ S 1 under the billiard flow before returning to the section M. As with holes of Type I, we define H σ to be the boundary of H σ viewed as a subset of M. The set B σ as a subset of M has similar geometric properties as f 1 H σ for Type I holes, i.e., f 1 ( H σ )\( M f 1 ( M)) consists of pairs of negatively sloped curves ending in M f 1 ( M). The slopes of these curves are uniformly bounded (independent of σ) away from and 0. For the reasons discussed, it will be convenient to view this set as part of the discontinuity set of f. The infinitesimal hole H 0 M is defined in the natural way, and the analog of Lemma 3.1 can be verified. We will say more about the geometry of H σ in Sect

18 Points q 0 that lie on multiple tangencies lead to slightly more complicated geometries, and special care is needed when defining what is meant by the sets f 1 H 0, H 0, and fh 0 as in the case of Type I holes. Further generalizations on holes of Type II: In addition to the generalizations discussed in Sect. 1.3, sufficient conditions on the holes allowed in Σ h for Prop. 2.2 to remain true are the following, as can be seen from our proofs: (1) There exist N and L for which the following hold for all sufficiently small h: (a) f 1 ( H σ ), H σ, and f( H σ ) each consist of no more than N smooth curves, all of which have length no greater than L. (b) For each σ Σ h, f 1 ( H σ )\( M f 1 ( M)) consists of piecewise smooth, negatively sloped curves (with slopes uniformly bounded away from and 0), and the end points of these curves must lie on M f 1 ( M). (2) The analog of Lemma 3.1 holds. Thus it would be permissible to allow a convex hole σ to be in Σ h that did not have a C 3 simple closed curve with strictly positive curvature as its boundary. For example, conditions (a) and (b) above hold if σ is a piecewise C 3 simple closed curve which consists of finitely many smooth segments that are either strictly positively curved or flat. As another generalization, consider the case when any line segment in the table X with its endpoints on two scatterers that passes through the convex hull of σ also intersects σ. Then it is no loss of generality to replace σ by its convex hull. Using this, one can often verify that the set H σ that arises satisfies properties (a) and (b) above, even if σ is not itself convex. See Fig. 3. or or Figure 3: Examples of Type II holes that are permissible. In Sect. 3.2, the discussion is for holes of Type I with a single interval deleted. The proof follows mutatis mutandis for holes of Type II, with the necessary minor modifications discussed in Sect Proof of Proposition 2.2 (for holes of Type I) The idea of the proof is as follows. First we construct a horseshoe (Λ (0), R (0) ) with the desired properties for the infinitesimal hole {q 0 }. Then we construct (Λ (σ), R (σ) ) for all σ Σ h (q 0 ), and show that with Λ (σ) sufficiently close to Λ (0) in a sense to be made precise, (Λ (σ), R (σ) ) will inherit the desired properties with essentially the same bounds. To ensure that Λ (σ) can be taken close enough to Λ (0), we decrease the size of the hole, i.e., we let h 0. Now 17